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I am further finalizing/completing/expanding the extended pdf slide notes that I had prepared for the Halifax CMS meeting last week (see another thread). I guess I will (re-)use them in September, in or around the Conference on Type Theory, Homotopy Theory and Univalent Foundations in Barcelona.
Today I have been working more on the section on
(somewhere around page 41/slide 59)
This is due to work that Joost Nuiten is currently doing with me, for his Master thesis. In that section I am trying to indicate
a detailed definition and analysis that we have for motivic quantization in 2d and how it holographically yields the quantization of Poisson manifolds, putting (we think) both the Kontsevich+Cattaneo-Felder as well as the Gukov-Witten story in a synthetic/axiomatic/conceptual context;
an idea of the general axiomatics of which this is a special case.
(Only that the slides go about this in opposite order.)
The second point is still a bit tentative, as I hope the slides make clear, but it seems that a clear picture is emerging.
Typo
the most intersting (p. 42)
Is there something ’chromatic’ about that $gl_1$ occuring on p. 44? I mean might a passage along the chromatic pattern be possible?
I seem to be obsessing on chromatic things at the moment. Perhaps physics is happy staying at $n = 1$.
Hi David,
thanks for catching the typo! Am fixing it now.
Concerning chromatics: hm, these two “GL”-s here are different. What I am talking about is the infinity-group of units of an $E_\infty$-ring spectrum. That $E_\infty$-ring might be $tmf$, for instance, hence be of chromatic level 2. (Maybe I shouldn’t write “$\mathbf{K}$” for the generic such $\infty$-rings in my notes, it can be misleading, I guess.)
So the next step to the story that you saw in the notes as an example for $n = 1$ will be the inclusion of circle 2-bundles into $tmf$ as indicated here
And that’s good, because physics is definitely not happy with staying at chromatic level 1! :-) We already know that $tmf$ at $n = 2$ controls the string. We just would like to understand it at a more deeper/more detailed level.
I think I’m getting a little confused as to why all those branes of the bouquet already appear by the time of super cohesion (here). Will there be new branes for less truncated cohesion?
Hi David,
hm, here I am not sure what you mean. These GS super-branes are incarnations of extensions of super $L_\infinity$-algebras. So they “appear” as soon as the notion of super-$L_\infty$-algebra is available. Depending on how one sets things up, this is the case as soon as one has super-grading, infinitesimals and homotopy.
Does that help? Maybe I am not understanding your question yet.
I probably have this all wrong, but if super-branes appear
as soon as one has super-grading, infinitesimals and homotopy,
and if super-grading arises from a 2-truncation, shouldn’t there be something analogous to super-branes at higher truncations?
From tmf:
One of the greatest recent achievements in algebraic topology is the construction of the tmf spectrum as the global section of a certain (infinity,1)-sheaf of commutative ring spectra over the moduli stack of elliptic curves.
I am not sure if the $n$Lab writers are aware that this result of Jacob Lurie has been conjectured by Gabriele Vezzosi in 2002. Maybe there is some place to acknowledge this fact.
@David,
ah, now I see, you are of course still after that super-general picture of supergeometry. As usual, you are pushing for the next major goal that has appeared far at the horizon of the conceptual general abstract.
So, to my shame I have to admit that I still don’t know the answer to your question then. Currently I am looking at what 99 per cent of the community seem to regard as a fairly abstract perspective on supergeometry, namely just the geometry in the topos over superpoints. But you are saying: wait, this is just a faint shadow of the full supergeometry, which should be the geometry under $Spec \mathbb{S}$. What happens to the notion of super-$p$-branes when regarded from this more general perspective?
Alas, I don’t know. But I fully agree that somebody should try to push further in this direction.
@Zoran,
the canonical place to acknowledge this would seem to be that very entry on tmf.
May I suggest that we take this discussion to another thread, though? It seems to have nothing specifically to do with the topic of this thread here.
@Urs,
I was just wondering if it was possible to glimpse how things might be. Rather like with Baez and Dolan and the cobordism hypothesis, sometimes a simple enough conception leads to a bold conjecture, even if it gives no indication of how to prove anything.
I was hoping that Joyal’s comment on stable cohesion might give us a clue. And then maybe physics has given an indication that it needs something more than super-branes.
I probably am guilty of something similar to Russell’s complaint
The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil.
Something like
The method of “speculating” without doing the hard groundwork has many advantages; they are the same as the advantages of taking the cherry off the top of the cake.
Sorry to harp on, but in what you added at super algebra,
All this suggests that in full generality supergeometry is to be thought of as S-graded geometry, hence geometry under Spec(S),…
why would you still want to call this supergeometry? Why not call it stable-geometry?
David,
I entirely agree! It’s just that at this point I don’t know of a good speculation to make.
Except for the generic speculation, which however might cause people to complain about serial robbery:
We see that in the perturbative/low-energy supergravity approximations the fundamental objects ($p$-branes) appear as canonical objects in “ordinary” supergeometric cohomology, using the grading implied by the low truncation of the sphere spectrum. So there are two truncations here that match:
Given this, the canonical crime to commit would be to conjecture that as we lift the truncation on the left we also lift the truncation on the right. This would be an M-theory capital offense…
I promise that as soon as a get hold of any further hint as to what is going on here, I’ll drop a note. Or else you should drop me a note, if you find out anything…
David,
okay, let me experiment.
So I suppose to capture this nicely abstractly, we want to say that we are in an ambient homotopy type theory context exhibiting differential cohesion and containing a good supply of stable objects, in Joyal’s sense.
It remains to say what the last clause means abstractly/axiomatically. Probably it just means to add the axiom that asserts the existence of one stable object. But I need to check/think.
Probably we are going to call these axioms “stable differential cohesion” or the like.
So then in such a context, we have at our hands the following two notions:
we have a notion structure sheaves $\mathcal{O}_X$ for each object $X$, as discussed at differential cohesion in the section on structure sheaves.
by stability we should be able to say that there is an object $\mathbb{S}$ (or rather $\mathbb{S} \to \ast$, the geometric sphere spectrum regarded as a bundle of spectra over the point).
Then we should also be able to speak of the constant sheaf $const_X \mathbb{S}$ of rings on $X$. (Hm, this sounds trivial, but I need to think about it in this context…). If so, a “super-general-super-grading” on the structure sheaf of $X$ would be a map
$\mathcal{O}_X \to const_X \mathbb{S}$of finite $\infty$-limit preserving $\infty$-functors $\mathbf{H} \to Sh_{\mathbf{H}}(X)$.
To make this more concrete, if we do happen to be explicitly in an $E_\infty$-algebraic model, then this should essentially just mean that a map $Spec \mathbb{S} \to X$ which should exhibit $X$ as being “super-generally-supergeometric”, is witnessed via Yoneda by turning any $A$-valued function $X \to A$ into a function $Spec \mathbb{S} \to A$ (by precomposition).
Er, hm, or maybe not. I’d need to think more.
Is there something to learn from TAF not relating to higher dimensional super-QFT when it reaches beyond chromatic level 2?
There’s a table on p. 75 of Behrens’ talk 1 which has him asking what represents cocycles for TAF, and at the same time asking for the geometry that follows ’spin’ and ’string’. Can’t say that I can see his answers from the later talks.
Why spin/string as chromatic levels 1/2 of Behrens’ table and spin/string as 2/3 of the table at super algebra? What would chromatic level have to do with truncation?
re #15:
So the cotangent complex construction $\Omega$ via tangent $\infty$-categories is
$(\Omega \dashv dom) \;\colon\; T_C \stackrel{\overset{\Sigma^\infty_C}{\leftarrow}}{\underset{\Omega^\infty_{C}}{\to}} C^{\Delta^1} \stackrel{\overset{const}{\leftarrow}}{\underset{dom}{\to}} C \colon \Omega \,.$If we want to connect this to cohesive $\infty$-toposes, we need to identify $C = \mathbf{H}^{op}$, since $C$ here is to be an $\infty$-category of algebras, not of spaces.
So we get
$(T_{\mathbf{H}^{op}})^{op} \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} ((\mathbf{H}^{op})^{\Delta^1})^{op} \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} \mathbf{H} \colon \Omega \,.$So…
[ woops, I just see that I need to run. Message to be continued later… sorry. ]
Let’s continue discussion this particular issue here.
of as S-graded geometry, hence geometry under Spec(S)
You mean over $Spec(S)$ (geometrically you are in overcategory, not undercategory, unlike in dual language of superalgebras).
No, that’s the thing, it’s under.
Geometry under $Spec \mathbb{S}$ is algebra over $\mathbb{S}$ and this is $\mathbb{S}$-graded algebra.
While the variance is right, I do agree that something else is wrong, concerning the relation to superalgebra. It’s not actually that we want maps of $E_\infty$-algebras $A \to \mathbb{S}$. I think we rather want maps from the corresponding infinity-group of units to $\mathbb{S}$ regarded just as an abelian $\infty$-group, and actually I think we may want
$gl_1(A) \to \mathbb{S} \,.$This would bring this here into the game:
But I need to interrupt now for a telephone call..
Yes, that’s it. See also Sagave’s section 1.4
Hm, nice. So I thought this superalgebra discussion here was off-topic for this thread on “motivic quantization”, but now we are coming back full circle to that topic and both aspects seem to nicely unify.
Okay, I think I’ve got. Actually.
To make notes somewhere, I ended up making them in the pertinent section superalgebra - Abstract idea.
What I put there now uses a bit of stuff as indicated in my talk notes, but details of which are currently behind-the-scenes. I have added a disclaimer behind the above link which aims to make this status clear. If anyone feels that this is too mysterious to be public on the nLab, then please complain and I’ll remove this until the background material is out (due out end of August, thesis by my student Joost Nuiten).
Anyway. Based on this I think I know now what the proper name of the next non-vanishing homotopy group of the sphere, after
is. It is (and this may be a shock now): Yang monopole.
The one after that should then indeed be fivebrane, but to be sure about that I am still missing one input. Maybe I’ll find it after a little thinking…
Just back from a conference meal, so not feeling too sharp, but now I am very confused.
Over the last few days, sometimes we seem to be dealing with the chromatic story and sometimes with homotopic truncation, e.g as mentioned above where the spin/string sequence is used for both.
And then there’s this tangent category to a cohesive topos business, where it might seem that there are stable forms of cohesion to match the existing forms. But then what is stable super-cohesion, when I was hoping that ’super’ always appears when a fuller story is truncated as in super algebra.
So I was wondering if there’s really some proper categorification going on in a way that the stability introduced by the tangent category did not, and I see now you have super 2-algebra, which has me thinking that what I was really after was some kind of $\infty$-algebra.
I think I need to get to bed.
Let me try to briefly clear it up, since it seems to be really a rather nice picture. But I, too, should call it quits, so maybe we postpose a more substantial exchange until tomorrow.
But here is the punchline, to my mind:
tangent $\infty$-toposes seems to be the way to include stable homotopy theory in cohesive homotopy theory.
commutative algebra in stable homotopy theory is automatically and intrinsically higher supergeometric: the rings of units of any commutative $\infty$-ring are automatically super-rings, in the higher sense, as Kapranov hinted at;
(I am thinking that this “explains” neatly why there is this abundance of models of K-theory using supergeometry, and at least a shadow of a model of tmf also using supergeometry (fermionic nets). )
So in a way your question for “stable super-cohesion” is making it more complicated than it is. Stable commutative algebra already brings superalgebra with it. That’s the remarkable punchline of combining Kapranov’s observation with Sagave’s.
So the Yang instanton (said page doesn’t exist, by the way) lives on $S^5$, much as the Yang monopole lives on $S^4$?
So in a way your question for “stable super-cohesion” is making it more complicated than it is. Stable commutative algebra already brings superalgebra with it.
That’s what I thought would happen, but still there is a tangent category T(Super∞Grpd) which presumably is different from T(∞Grpd), and were it also to be an $\infty$-topos, there would then be a stable super-cohesion to go along with stable cohesion. Or perhaps you’re predicting they’re equivalent. Or maybe T(Super∞Grpd) isn’t an $\infty$-topos.
@David R, re #25
First, I do mean the Yang monopole of course (there is no “Yang instanton”), I have fixed it now in the above, but in the entry section it had been correct all along.
Second, the Yang monopole is, as a brane, a 4-brane with 5 = 4+1 dimensional worldvolume.
To make this clear, I have now added a good bit of text to Yang monopole. Have a look!
@David C, re #26:
You are absolutely right with your questions here. I cannot guarantee that I already see everything of the full picture, but this here is how I currently view it:
Down in $T (\infty Grpd)$ or rather in $CAlg( T(\infty Grpd), \otimes )$ Kapranov-style higher supergeometry already exists canonically.
Now, as we consider motivic/cohomological quantization of $\sigma$-models in some cohesive $\mathbf{H}$, we are to find maps
$\mathbf{B} n Line \to B gl_1(E)$from the coefficients $n Line$ of our local action functionals (for instance $n Line = \mathbf{B}^n U(1)$ in the archetypical case) to the delooped $\infty$-group of units of the $E_\infty$-ring $E$ which we take to be the “linearization” of the coefficients.
For instance for quantizing the particle on the boundary of the string, we may use the map
$B BU(1) \to B gl_1(KU) \,.$That this exists is the fact that complex K-theory has a twist by degree-3 classes. But KU has more twists, meaning that $B gl_1(KU)$ is bigger than the image of the above map . By Kapranov+Sagave we know that the twists arrange into a higher supergeometric structure. But of course for the example of complex K-theory this is a standard fact: the full twists are induced not from the ordinary lines classified by BU(1), but by the super-lines.
This is discussed in some detail at line 2-bundle – Super line 2-bundles and twisted K-theory.
So if we are in $\mathbf{H} = Smooth \infty Grpd$ and look at $KU$ there, then $KU$ itself knows that it is a supergeometric coefficient for motivic/cohomological quantization, but the local action functionals in $\mathbf{H}$ are not rich enough to probe all that supergeometric freedom in $KU$.
But as we pass to $\mathbf{H} = SmoothSuper \infty Grpd$ the local action functionals do become rich enough to map to all the available “quantum twists”.
By extrapolation I’d expect (but haven’t checked) that if we take some higher cohomology theory such as $tmf$ or some $\tilde K(n)$, then similarly in order for local action functionals to exhaust the available freedom we need a suitably higher-Kapranov-style supergeometric ambient $\infty$-topos.
In summary I think what I am saying is simply this:
$CAlg(T (\infty Grpd), \otimes)$ already has built in higher super-algebra structure;
but to cohesively refine this we may need very richly higher-supergeometric ambient $\infty$-toposes.
That’s at least how I would currently think about it.
But KU has more twists, meaning that $Bgl_1(MU)$
What’s the relationship between KU and MU?
At super algebra this confused me too where it says
To see how ordinary superalgebra arises this way, consider the case of KU.
and goes on to talk about MU.
On another point, do you envisage the need after ’line $n$-bundle’ and ’super line $n$-bundle’ for ’spin line $n$-bundle’, ’string line $n$-bundle’, etc.? What would the limit case be called? ’Stable line $n$-bundle’? But I wonder if there’s a kind of double stabilization going on
T(∞Grpd), T(Super∞Grpd), T(Spin∞Grpd), T(String∞Grpd),…
On yet another note, what is there to be said about iterating the T construction? Is T(T(∞Grpd)) interesting? It seems like there’s some interesting structure in iterated tangent bundles from this MO question.
Hi David,
concerning your first point: ah, that’s emberrassing, these MUs are typos introduced by my fingers. They should all be KUs. I have fixed it now.
Concerning your second: yes, that’s the next question, exactly. If we say we introduce ordinary supergeometry such as to be able to obtain all the twists of complex K-theory from geometry, then the next question is which higher kind of supergeometry we’d need in order to obtain all twists of tmf geometrically. Naturally, at this point I don’t know, but this is exacly how I am thinking one should approach this idea of “Kapranov higher supergeometry” now.
Concerning your third point, on iteratated tangent $\infty$-categories: hm, I don’t know. You clearly are coming up with good questions faster than I can think up good answers :-)
“Kapranov higher supergeometry”
It’s just a name, but I’m still not totally convinced by the ’higher super’ tag. Isn’t that a little like saying of triple and quadruple that they’re ’higher doubles’? Or of the natural numbers that they’re ’higher twos’?
Yes, we need a better word, true. Maybe “homotopy supersymmetry” or $\infty$-supersymmetry. Well, or just “$\mathbb{S}$-grading” :-)
Shouldn’t we drop any mention of ’super’ from the big picture since it’s just a sign of 2-truncation?
Hmm, could that thought possibly relate to that idea above about
TAF not relating to higher dimensional super-QFT ?
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